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An Empirical Characteristic Function Approach to VaR under a Mixture of Normal Distribution with Time-Varying Volatility

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Author Info

  • Dinghai Xu

    (Department of Economics, University of Waterloo)

  • Tony S. Wirjanto

    (Department of Economics, University of Waterloo)

Abstract

This paper considers Value at Risk measures constructed under a discrete mixture of normal distribution on the innovations with time-varying volatility, or MN-GARCH, model. We adopt an approach based on the continuous empirical characteristic function to estimate the param eters of the model using several daily foreign exchange rates' return data. This approach has several advantages as a method for estimating the MN-GARCH model. In particular, under certain weighting measures, a closed form objective distance function for estimation is obtained. This reduces the computational burden considerably. In addition, the characteristic function, unlike its likelihood function counterpart, is always uniformly bounded over parameter space due to the Fourier transformation. To evaluate the VaR estimates obtained from alternative specifications, we construct several measures, such as the number of violations, the average size of violations, the sum square of violations and the expected size of violations. Based on these measures, we find that the VaR measures obtained from the MN-GARCH model outperform those obtained from other competing models.

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Bibliographic Info

Paper provided by University of Waterloo, Department of Economics in its series Working Papers with number 08008.

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Date of creation: Dec 2008
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Handle: RePEc:wat:wpaper:08008

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Related research

Keywords: Value at Risk; Mixture of Normals; GARCH; Characteristic Function.;

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References

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  1. Vlaar, Peter J G & Palm, Franz C, 1993. "The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 11(3), pages 351-60, July.
  2. Luc Bauwens & Charles S. Bos & Herman K. van Dijk, 1999. "Adaptive Polar Sampling with an Application to a Bayes Measure of Value-at-Risk," Tinbergen Institute Discussion Papers, Tinbergen Institute 99-082/4, Tinbergen Institute.
  3. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, Econometric Society, vol. 50(4), pages 987-1007, July.
  4. Haas, Markus & Mittnik, Stefan & Paolella, Marc S., 2002. "Mixed normal conditional heteroskedasticity," CFS Working Paper Series, Center for Financial Studies (CFS) 2002/10, Center for Financial Studies (CFS).
  5. Bai, Xuezheng & Russell, Jeffrey R. & Tiao, George C., 2003. "Kurtosis of GARCH and stochastic volatility models with non-normal innovations," Journal of Econometrics, Elsevier, Elsevier, vol. 114(2), pages 349-360, June.
  6. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 347-70, March.
  7. Knight, John L. & Yu, Jun, 2002. "Empirical Characteristic Function In Time Series Estimation," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 18(03), pages 691-721, June.
  8. Gray, Stephen F., 1996. "Modeling the conditional distribution of interest rates as a regime-switching process," Journal of Financial Economics, Elsevier, Elsevier, vol. 42(1), pages 27-62, September.
  9. Christoffersen, Peter F, 1998. "Evaluating Interval Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 841-62, November.
  10. Jose Lopez, 1998. "Methods for evaluating value-at-risk estimates," Research Paper, Federal Reserve Bank of New York 9802, Federal Reserve Bank of New York.
  11. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  12. repec:fth:louvco:9957 is not listed on IDEAS
  13. Schwert, G William, 1989. " Why Does Stock Market Volatility Change over Time?," Journal of Finance, American Finance Association, American Finance Association, vol. 44(5), pages 1115-53, December.
  14. Hamilton, James D. & Susmel, Raul, 1994. "Autoregressive conditional heteroskedasticity and changes in regime," Journal of Econometrics, Elsevier, Elsevier, vol. 64(1-2), pages 307-333.
  15. Franc Klaassen, 2002. "Improving GARCH volatility forecasts with regime-switching GARCH," Empirical Economics, Springer, Springer, vol. 27(2), pages 363-394.
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Cited by:
  1. Dinghai Xu, 2009. "The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey," Working Papers, University of Waterloo, Department of Economics 0904, University of Waterloo, Department of Economics, revised Sep 2009.
  2. Tony S. Wirjanto & Adam W. Kolkiewicz & Zhongxian Men, 2013. "Stochastic Conditional Duration Models with Mixture Processes," Working Paper Series, The Rimini Centre for Economic Analysis 29_13, The Rimini Centre for Economic Analysis.

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